Question

Evaluate fx, fy, fz at the given point a) f (x, y z) = x³yz² at the point (1, 2, 3) b) f (x, y, z) = x² - 2xy + 3yxz² at the point (3, 1, -2)

Answer 1

Answer:

• a) (fx, fy, fz) = (54, 9, 12)
• b) (fx, fy, fz) = (16, 30, 9)

Step-by-step explanation:

a) The partial derivatives of f(x, y, z) = x³yz² are ...

• fx = 3x²yz²
• fy = x³z²
• fz = 2x³yz

At the given point, these are ...

• fx(1, 2, 3) = 3(1²)(2)(3²) = 54
• fy(1, 2, 3) = (1³)(3²) = 9
• fz(1, 2, 3) = 2(1³)(2)(3) = 12

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b) The partial derivatives of f(x, y, z) = x² -2xy +3xyz² are ...

• fx = 2x -2y +3yz²
• fy = -2x +3xz²
• fz = 3xy

At the given point, these are ...

• fx(3, 1, -2) = 2(3) -2(1) +3(1)(-2)² = 16
• fy(3, 1, -2) = -2(3) +3(3)(-2)² = 30
• fz(3, 1, -2) = 3(3)(1) = 9